Rose (mathematics)

Rose with 7 petals (k = 7)

Rose with 8 petals (k=4).

Rose curves defined by

r=\cos k\theta

, for various values of k=n/d.

In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates.

General overview[edit]

Up to similarity, these curves can all be expressed by a polar equation of the form

\!\,r=\cos(k\theta )

^[1]

or, alternatively, as a pair of Cartesian parametric equations of the form

\!\,x=\cos(k\theta )\cos(\theta )

\!\,y=\cos(k\theta )\sin(\theta )

If k is an integer, the curve will be rose-shaped with

2k petals if k is even, and
k petals if k is odd.

Where k is even, the entire graph of the rose will be traced out exactly once when the value of theta, θ changes from 0 to 2π; when k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for k even, and π for k odd.)

If k is a half-integer (e.g. 1/2, 3/2, 5/2), the curve will be rose-shaped with 4k petals. Example: n=7, d=2, k= n/d =3.5, as θ changes from 0 to 4π.

If k can be expressed as n±1/6, where n is a nonzero integer, the curve will be rose-shaped with 12k petals.

If k can be expressed as n/3, where n is an integer not divisible by 3, the curve will be rose-shaped with n petals if n is odd and 2n petals if n is even.

If k is rational, then the curve is closed and has finite length. If k is irrational, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set (i.e., it comes arbitrarily close to every point in the unit disk).

Since

\sin(k\theta )=\cos \left(k\theta -{\frac {\pi }{2}}\right)=\cos \left(k\left(\theta -{\frac {\pi }{2k}}\right)\right)

for all $\theta$ , the curves given by the polar equations

\,r=\sin(k\theta )

and

\,r=\cos(k\theta )

are identical except for a rotation of π/2k radians.

Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728.^[2]

Area[edit]

A rose whose polar equation is of the form

r=a\cos(k\theta )\,

where k is a positive integer, has area

{\frac {1}{2}}\int _{0}^{2\pi }(a\cos(k\theta ))^{2}\,d\theta ={\frac {a^{2}}{2}}\left(\pi +{\frac {\sin(4k\pi )}{4k}}\right)={\frac {\pi a^{2}}{2}}

if k is even, and

{\frac {1}{2}}\int _{0}^{\pi }(a\cos(k\theta ))^{2}\,d\theta ={\frac {a^{2}}{2}}\left({\frac {\pi }{2}}+{\frac {\sin(2k\pi )}{4k}}\right)={\frac {\pi a^{2}}{4}}

if k is odd.

The same applies to roses with polar equations of the form

r=a\sin(k\theta )\,

since the graphs of these are just rigid rotations of the roses defined using the cosine.

How the parameter k affects shapes[edit]

In the form k = n, for integer n, the shape will appear similar to a flower. If n is odd half of these will overlap, forming a flower with n petals. However, if it is even the petals will not overlap, forming a flower with 2n petals.

When d is a prime number then n/d is a least common form and the petals will stretch around to overlap other petals; the number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.

In the form k = 1/d when d is even then it will appear as a series of d/2 loops that meet at 2 small loops at the center touching (0, 0) from the vertical and is symmetrical about the x-axis. If d is odd then it will have d div 2 loops that meet at a small loop at the center from either the left (when in the form d = 4n − 1) or the right (d = 4n + 1).

If d is not prime and n is not 1, then it will appear as a series of interlocking loops.

If k is an irrational number (e.g. $\pi$ , ${\sqrt {2}}$ , etc.) then the curve will have infinitely many petals, and it will be dense in the unit disc.

Examples of roses created using gears with different ratios

n=1, d=1, k=1.

n=1, d=3, k≈0.333

n=3, d=1, k=3

n=4, d=5, k=0.8

Offset parameter[edit]

Play media

Animating effect of changing offset parameter

Adding an offset parameter c, so the polar equation becomes

\!\,r=\cos(k\theta )+c

alters the shape as illustrated at right. In the case where the parameter k is an odd integer, the two overlapping halves of the curve separate as the offset changes from zero.

Programming[edit]

R

k <- 4
t <- seq(0, 4*pi, length.out=500)
x <- cos(k*t)*cos(t)
y <- cos(k*t)*sin(t)
plot(x,y, type="l", col="blue")

MATLAB and OCTAVE

function rose(del_theta, k, amplitude)
% inputs:
%   del_theta = del_theta is the discrete step size for discretizing the continuous range of angles from 0 to 2*pi
%   k = petal coefficient
%      if k is odd then k is the number of petals
%      if k is even then k is half the number of petals
%   amplitude = length of each petal
% outputs:
%   a 2D plot from calling this function illustrates an example of trigonometry and 2D Cartesian plotting
theta = 0:del_theta:2*pi;
x = amplitude*cos(k*theta).*cos(theta);
y = amplitude*cos(k*theta).*sin(theta);
plot(x,y)